Biomedical Engineering Reference

In-Depth Information

temperature.
42
In our studies of CBE in images, apparent motion

of image features has been tracked and compensated for auto-

matically as described previously, so that CBE can be measured

at each pixel in motion-compensated images.
70
Figure 13.8 shows

CBE images, that is, the energy changes relative to the reference

image at 37°C. As predicted, the energy change is both positive

(increasing) and negative (decreasing) with temperature. This

approach allows use of the whole ultrasonic image rather than

just the signals from selected scattering regions, but at the price

of possibly increasing noise in estimates of CBE.

After compensating for apparent motion in images of bovine

liver, turkey breast, and pork muscle, the mean change in back-

scattered energy at each pixel over eight image regions in all tissue

specimens was calculated with respect to a reference temperature

(37°C). As temperature increased, for some scattering regions the

CBE was positive, for others it was negative as seen in Figure 13.9

for our initial predictions, from simulations of images of scatterer

populations, and from
in vitro
measurements in different types

of tissue. Because the means of CBE from pixels with positive

and negative relative backscattered energy changed nearly mono-

tonically, CBE is a suitable parameter for temperature estimation.

From uniform-heating studies its accuracy and spatial resolution

appear to be suitable for temperature imaging.
73

CBE reference (37°C)

CBE @ 41°C

dB

10

10

2

2

0

0

4

4

6

−10

6

−10

5101520

5101520

CBE @ 45°C

CBE @ 50°C

10

10

2

2

0

0

4

4

6

6

−10

−10

5101520

5101520

mm

mm

FIGURE 13.8
Change in backscattered energy in ultrasound images

of bovine liver from 37 to 50°C after compensation for apparent

motion. All images were referred, pixel-by-pixel, to the energy in the

reference image at 37°C. Each color bar is in dB (similar to Figure 6,

but from a different specimen and over a larger region of interest, in

Arthur et al.
43
).

where, as functions of temperature, α(
T
) is the attenuation within

the tissue volume and η(
T
) is the backscatter coefficient of the

tissue volume. Distance
x
is the path length in the tissue volume.

We inferred the temperature dependence of the backscatter

coefficient from the scattering cross section of a sub-wavelength

scatterer.
114,115
Neglecting the effect of the small change in the

wavenumber (<1.5%) with temperature, we assumed
91

13.4.4.1 Stochastic-Signal Framework

In our initial work, CBE from backscattered signals was com-

puted as a ratio at each pixel in the envelope-detected images of

energies at temperature
T
and
T
R
.
43
It was characterized by aver-

aging ratios larger than and less than 1, denoted as positive CBE

(PCBE) and negative CBE (NCBE), which describe the increase

and decrease in the backscattered energy, respectively.

When
i
en
is represented by a random process, computation of

the signal ratio can be modeled as a ratio between two random

variables,
y
T
and
y
R
:
46

2

2

2

2

ρ

cT

()

−ρ

cT

()

1

3

33

2

ρ−ρ

ρ+ρ

mms

s

s

m

+

2

ρ

cT

()

(13.8)

η

η

()

()

T

T

s

s

s

m

=

,

2

2

2

2

ρ

cT

()

()

()

−ρ

cT

1

3

33

2

ρ−ρ

ρ+ρ

R

mRms

Rs

s

m

+

y

y

z

=

T

R

(13.9)

2

ρ

cT

s

Rs

s

m

where ρ and
c
are the density and speed of sound of the scatterer

s
and medium
m
. This expression applies to conditions where

the wavelength λ is larger than 2π
a
, where
a
is the radius of the

scatterer. Assuming a speed of sound of 1.5 mm/μs and a fre-

quency of 7.5 MHz, λ is 0.2 mm, which means that Equation 13.7

using Equation 13.8 applies to scatterers smaller than 30 μm.

Changes in backscattered energy were modeled assuming that

the scattering potential of the volume was proportional to the

scattering cross section of sub-wavelength scatterers. We pre-

dicted with this model that the change in backscattered energy

could increase or decrease depending on what type of inhomo-

geneity caused the scattering. These calculations suggested that

the change in backscattered energy could vary depending on the

type of scatterers in a given tissue region.

In 1D studies, we showed that it is possible to isolate and mea-

sure backscattered energy from individual scattering regions,

and that measured CBE was nearly monotonically dependent on

where
y
R
and
y
T
are the random variables representing the

B-scans at the reference and current temperatures. The ratio,

z
, is also a random variable whose probability density function

(PDF),
f
Z
(
z
), is determined by the joint distribution of (
y
R
,
y
T
):
116

∞

∫

fz

()

=

||

yf

(,

yyzdy

)

,

(13.10)

Z

R

YY

RR

R

RT

−∞

where
y
R
,
y
T
, and
z
> 0.

The computation of PCBE in our initial work can be written

as

1

∑

z

N

k

1

∑

kkz

N

N

∈>

{|

1}

PCBE

=

z

=

,

k

k

N

+

+

kkz

∈>

{|

1}

k

where
N
is the number of pixels in image and
N
+
is the number of

ratio pixels with value larger than 1. Assuming
z
k
's are independent,